![there are two possible outcomes. Since we have two possible outcomes, one possible outcome is state
0, the other possible outcome is state 1. This means that each possible outcome is 1/2. We say that
each possible outcome is a chance and the chance for a possible outcome is 1/2. In other words, the
chances are 1/2 that it will be in a state.
We take probability a step further with something called probability amplitudes which we refer
to as just amplitudes. An amplitude is a complex number meaning it has a magnitude and phase. The
phase is an angle. I will just refer to the magnitude of the amplitude as the amplitude since we can’t
measure the phase and therefore can’t use the phase. This is also a standard reference, or consensus, in
the quantum computing field. The amplitude is just the square root of the chance. So, if the probability
of a state is 1/2 then the amplitude is √
1
2
. This also means that the chance or probability is the squared
of the amplitude. So, for example, if the amplitude is 1/4 then the probability is 0.252
= 1/16.
Don’t let this confuse you. If you have the amplitude square it to get the probability and if you
have the probability square root it to get the amplitude. The range of probability is from 0 to 1. The
range of amplitude is from -1 to 1. The amplitude can be -1 because (−1)2
= 1, which is within range of
probability. However, the total probability of all the states must equal to 1. So, the probability of state 0
can be 1/2 and the probability of state 1 can be 1/2. Adding the two states together 1/2 + 1/2 =1. If the
amplitude of both states is 1/4 then it is not valid in quantum computing because (1/4)2
= 1/16.
Thus, the total probability would be 1/16 + 1/16 = 1/8. Or if the amplitude is 1 for both states, then the
total probability is 12
+ 12
= 2 which is also not valid.
Vectors
Now, we can talk about vectors. A vector is a set of elements. We use vectors as another way to
list the amplitudes. It is just another representation. A state vector is the vector for a particular
quantum state of the qubits. The vector of state 0 is [
1
0
] which is called a state vector for state 0. This
says that the amplitude of state 0, which is on top, is 1 and the amplitude of state 1, which is on the
bottom, is 0. The total probability is 12
+0=1 which means it is valid. The state vector for state 1 is [
0
1
]
which means the amplitude of state 0 is 0 and the amplitude of state 1 is 1. Again, the total probability
is equal to 1. Also, a scalar multiplied by a vector gives us a vector. For example, scalar variable C
multiplied with the state vector for state 0, which is C|0> = 𝐶 [
1
0
] gives us [
𝐶 ∗ 1
𝐶 ∗ 0
] = [
𝐶
0
]. This means
state 0 has amplitude value of C and state 1 has amplitude value of 0. This is just an example and is not
valid in quantum computing because the total probability is less than 1 if C is not 1. If C = 1 only then is
it valid.
The state vector, mathematically, can be used as a n x 1 matrix to carry out computations. An nx1
matrix is n rows and one column. The reason is that the total probability must equal 1 is that the
operations of matrices have to be reversible which means a matrix has to be unitary. The notation for
each state vector is called a ket. The ket for state 0 is |0> and the ket for state 1 is |1>. A ket is part of
the bra-ket notation. A bra-ket notation is also known as Dirac notation. Paul Dirac was a physicist that
came up with the notation for quantum mechanics. Bra is a linear form of the vector, which is a 1xn
matrix, meaning one row and n columns. The way we obtain bra from ket is we perform the complex
conjugate transpose of the ket form. The representation of bra for state 0 is <0| and bra for state 1 is](https://image.slidesharecdn.com/projectdraft-230830221115-1ce53b0c/85/project-draft-pdf-6-320.jpg)
![Introduction to quantum mechanics and quantum computing (part 3)
Gates
In the first article I talked about quantum mechanics. In my second article I talked about
quantum computing. In this article I will talk about gates, two qubits and how two qubits look like.
We start this talk with the bloch sphere. This is a visual representation of a qubit. There are
three axis, the X-axis, Z-axis and Y-axis. We will focus on the X and Z axis because they are
straightforward. The Y-axis contains the imaginary plane. A picture of the bloch sphere is shown below
and will be referenced in this article and the rest of the series.
(Pic)
Usually in quantum computing, the qubits start out in state 0 like classical computing the bits
start at state 0. Quantum logical gates are called gates and allow us to perform operations
mathematically in quantum computing. So, an important and common gate is the X-gate it behaves like
a NOT in classical computing. It is a basic one qubit gate. An X-gate flips a bit from state 0 to state 1.
The notation for this transition is X|0> = |1>. This is because the position on the bloch sphere rotates
around the x axis. The matrix representation of a X-gate is [
0 1
1 0
]
hadamard, not, 2 qubit amplitude, cnot, more advanced gates are a combo other basic gates
mentioned, measurements, …..
I would like to thank the Womanium Global Quantum Computing Program for encouraging me to
start sharing my journey into the quantum world.
Below are resources, references and futher readings:
https://gitlab.com/qworld/bronze-qiskit
qiskit.org/learn/course/introduction-course
wikipedia
https://en.wikipedia.org/wiki/Quantum_computing
https://en.wikipedia.org/wiki/ Quantum_technology](https://image.slidesharecdn.com/projectdraft-230830221115-1ce53b0c/85/project-draft-pdf-8-320.jpg)
project draft.pdf
- 1. Introduction to quantum mechanics and quantum computing (part 1) Quantum Mechanics Quantum Mechanics is a branch of physics that deals with the world at the nanometer scale. In classical mechanics, which we usually call physics or classical physics, we are trying to understand the world at a larger scale above the nanometer scale. To contrast quantum and classical mechanics a nanometer is 1,000,000,000 times smaller than a meter or 0.000000001 meter. What is so special about the nanometer scale? It turns out that quantum mechanics exhibit different behaviors at that scale. The nanometer is not a hard line that defines the difference between classical and quantum mechanics. There is also modern physics that is a hybrid of the two types of mechanics. History Quantum mechanics started out in the early 19th century. However, the term quantum mechanics was coined in the early 20th century. In classical mechanics, around the 17th century, the famous Issac Newton developed a theory that light behaved like a particle. In the beginning of the 19th century a physicist named Thomas Young made a real-world experiment called the double-slit experiment to demonstrate that light acted like a wave. This experiment showed that when there were two slits on anything like a screen or something like a piece of paper then each particle of light passed through both slits at the same time and hit the screen behind it where there were no slits in the foreground and as if there was only one slit in the middle of the foreground screen (picture shown below). This showed similar results to single slit experiments where a single slit was in the middle of the screen or paper and the light hit the screen behind it exactly where the slit in the foreground was placed. This experiment proved the theory that light was a wave which was developed during the time of Issac Newton. However, Newton’s theory of light as a particle was more popular at that time. After this experiment and many others, statistical mechanics was developed. Statistical mechanics uses statistics to model randomness, uncertainty and such. (pic) In the early 20th century, two physicists Stern and Gerlach made an experiment called the Stern- Gerlach experiment. In this experiment they shot silver atoms between magnetics into a screen. The magnets showed the effects of a magnetic field on atoms. The atom passed two magnets, one north pole and one south pole. Regardless of how the atom was shot onto the screen but depending on the mass of the atom, the orientation of the magnets such as horizontal, vertical or at an angle and strength of the magnetic field that determined how it landed on the screen either on one side or the opposite side e.g., top or bottom, left or right. This is not to say that these factors controlled the atom in a particular direction. For example, the atom would go up or down on the screen in random order when those factors stayed the same, but we can’t control when they go up or when they go down. We don’t know the exact path that an atom takes but we do know where they end up which we can consider a measurement. This experiment also showed that particles have spins either spin up or spin down which are completely random, meaning they can’t be predicted or controlled. Later physicists
- 2. were able to develop on these experiments that showed that atoms remember their orientation and forget their orientation. We use these later experiments to develop quantum computers that we can measure the state and forget the state. We will talk about quantum computers in the next article. Theories About a half decade after the Stern-Gerlach experiment, in the mid to late 1920’s theories began to develop to understand or explain the phenomena that were shown in experiments for more than one century. Two top and contradicting theories were present until then. So, a question arose from them. Is light a particle or a wave? Complementarity that was developed by Niels Bohr that said we can measure to see if something like light to be a particle or a wave but not a particle and a wave at the same time. This is similar to superposition but not quite which we will talk about later. Another theory that built on the complementarity was called wave-particle duality. The wave-particle duality is a theory that was developed that explains that light has properties of both a particle and a wave. The property that shows the most depends on the environment and the action taken with light. It also holds that all particles behave like waves as we found out later. The Copenhagen interpretation were discussions surrounding many past experiments and their interpretations. It was about quantum mechanics in general not about a certain type of experiments or a particular phenomenon. A decade after the Copenhagen interpretation a hypothetical experiment called Schrodinger’s cat which was related to the Copenhagen interpretation. This was just an example not an actual experiment. Schrodinger’s cat said that if we put a cat in a box with a container filled with poison and the container breaks is the cat dead or alive? According to Schrodinger the cat was both dead and alive before looking inside the box. Before looking inside the box was considered something called superposition. Only after looking inside to see, we can determine if the cat is either dead or alive, but not both dead and alive. Looking inside the box was considered a measurement. Superposition is when an object, such as light, takes both paths like the double slit experiment. It can also be when something is doing both things at the same time. It can also be an object that has both states like Schrodinger’s cat. Superposition is used in quantum computing as two states simultaneously. Also, objects can be anything such as atoms or electrons as well. An object doesn’t break in pieces, takes all paths then become whole again. The object as a whole takes all paths at the same time. There is only a certain amount of chance to observe or measure the object in each state at a point in time. For example, there is a 1/2 chance that an object will be one state or the other state. How this happens is unknown because we can’t measure all the paths at the same time, that is considered an observation or measurement. When we try to observe all paths or states, superposition disappears or collapses into a single state or path. Since an object can take all paths, the path of the object is indeterministic because there is no way to know the current path without observing or measuring. When we observe then it becomes deterministic. In the early 20th century, a physicist named Werner Heisenberg creates a mathematical model that describes the limit of how much we know about an object. This is called the Heisenberg uncertainty principle. It shows if we know about the position of an object, we can’t know its momentum. For
- 3. example, if we know the position of an electron around the atom, we can’t know the momentum of the electron moving around the atom. Another example, if we know the momentum of an atom, we can’t know its location. We can’t know both at the same time and they have an inverse relationship. In the mid 20th century a physicist named John Wheeler wondered if we can decide to observe or not an object after it has taken both paths? This was called Wheeler’s delayed choice experiment which was just a bunch of hypothetical experiments. However, later we found out through other experiments, that we can actually do that. Wheeler’s delayed choice experiment would let us choose the experiment setup after the object passes both paths but before we observe it. He also wondered would the object know the choice we made after it passes through? The consensus is no but either way is hard to prove through experimentation. Each hypothetical experiment was posed from a question. In my next article we will talk about quantum computing and show how the theoretical and experimental fields of quantum mechanics have influenced the development of real-world quantum computers. I would like to thank the Womanium Global Quantum Computing Program for encouraging me to start sharing my journey into the quantum world. Below are resources, references and futher readings: edx https://www.edx.org/learn/quantum-physics-mechanics/georgetown-university-quantum-mechanics- for-everyone wikipedia https://en.wikipedia.org/wiki/History_of_quantum_mechanics https://en.wikipedia.org/wiki/Quantum_mechanics https://en.wikipedia.org/wiki/Timeline_of_quantum_mechanics https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics https://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment https://en.wikipedia.org/wiki/Schr%C3%B6dinger%27s_cat https://en.wikipedia.org/wiki/Double-slit_experiment https://en.wikipedia.org/wiki/Wave-particle_duality https://en.wikipedia.org/wiki/Copenhagen_interpretation
- 4. https://en.wikipedia.org/wiki/Wheeler's_delayed-choice_experiment https://en.wikipedia.org/wiki/Uncertainty_principle book: Introduction to Quantum Mechanics by David Griffiths (textbook also referred to as Griffiths) lectures: www.feynmanlectures.caltech.edu
- 5. Introduction to quantum mechanics and quantum computing (part 2) Implementations of Quantum Mechanics Quantum Computing My previous article was on quantum mechanics. Now let us talk about the implementations of quantum mechanics. Quantum computers have been built since the late 1990’s. In building a quantum computer it is consensus to use DiVincenzo’s Criteria much like classical computers use Moore’s Law. There are many types of quantum computer hardware. Further development of the Stern-Gerlach experiments helped develop quantum computers that can measure the state and forget the state. Even though there are different types of quantum computers they use quantum computing which is more software. Quantum computing is different than classical computing in that it uses the phenomena of quantum mechanics to compute. This is important to understand. If there is no quantum mechanics phenomena happening then it is not quantum computing. Quantum computing has similar models, methods, or protocols to compute across quantum hardware. The only difference is the implementation or code from one hardware type to the other. For example, quantum computing has algorithms such as Shor’s algorithm, Grover’s algorithm, Deutsch-Jozsa algorithm and many others. Quantum computing also uses different quantum error correction protocols just like classical computing uses error correction protocols. Also, just like classical computing has encryption, quantum computing also has quantum cryptography called quantum key distribution (QKD). We also have quantum information. Quantum information is a subset field of quantum computing in that quantum information is information about the state of a qubit and uses information theory or in this case quantum information theory. It is like software engineering is a subset field of computer science. Qubit Let me introduce to you the basics of quantum computing. First, we have a qubit. A qubit is a quantum bit. It can be the state 0, the state 1 or the indeterminate state of 0 and 1. These are the quantum states. In classical computing terms we have only two states, either state 0 or state 1. The indeterminate state of 0 and 1 is used as superposition. I have mentioned before in my previous article on quantum mechanics the term superposition. The indeterminate state of 0 and 1 is an unknown state until it is measured. We don’t know what the state will be while it is computed. Therefore, we don’t know the result of the computation until the measurement. A measurement in quantum computing is an observation. Superposition in quantum computing is a state that lets us have a quantum advantage over classical computing. Mathematics, Statistics and Superposition To understand how superposition works in quantum computing we must introduce basic statistics, specifically probabilities. Probability is important in quantum mechanics to explain phenomena but here I will show you how it is important in quantum computing. There is even something called statistical mechanics that is used. It is a field in statistics that is applied in quantum mechanics, but we won’t cover that here. Since there is an indeterminate state of 0 and 1 that means
- 6. there are two possible outcomes. Since we have two possible outcomes, one possible outcome is state 0, the other possible outcome is state 1. This means that each possible outcome is 1/2. We say that each possible outcome is a chance and the chance for a possible outcome is 1/2. In other words, the chances are 1/2 that it will be in a state. We take probability a step further with something called probability amplitudes which we refer to as just amplitudes. An amplitude is a complex number meaning it has a magnitude and phase. The phase is an angle. I will just refer to the magnitude of the amplitude as the amplitude since we can’t measure the phase and therefore can’t use the phase. This is also a standard reference, or consensus, in the quantum computing field. The amplitude is just the square root of the chance. So, if the probability of a state is 1/2 then the amplitude is √ 1 2 . This also means that the chance or probability is the squared of the amplitude. So, for example, if the amplitude is 1/4 then the probability is 0.252 = 1/16. Don’t let this confuse you. If you have the amplitude square it to get the probability and if you have the probability square root it to get the amplitude. The range of probability is from 0 to 1. The range of amplitude is from -1 to 1. The amplitude can be -1 because (−1)2 = 1, which is within range of probability. However, the total probability of all the states must equal to 1. So, the probability of state 0 can be 1/2 and the probability of state 1 can be 1/2. Adding the two states together 1/2 + 1/2 =1. If the amplitude of both states is 1/4 then it is not valid in quantum computing because (1/4)2 = 1/16. Thus, the total probability would be 1/16 + 1/16 = 1/8. Or if the amplitude is 1 for both states, then the total probability is 12 + 12 = 2 which is also not valid. Vectors Now, we can talk about vectors. A vector is a set of elements. We use vectors as another way to list the amplitudes. It is just another representation. A state vector is the vector for a particular quantum state of the qubits. The vector of state 0 is [ 1 0 ] which is called a state vector for state 0. This says that the amplitude of state 0, which is on top, is 1 and the amplitude of state 1, which is on the bottom, is 0. The total probability is 12 +0=1 which means it is valid. The state vector for state 1 is [ 0 1 ] which means the amplitude of state 0 is 0 and the amplitude of state 1 is 1. Again, the total probability is equal to 1. Also, a scalar multiplied by a vector gives us a vector. For example, scalar variable C multiplied with the state vector for state 0, which is C|0> = 𝐶 [ 1 0 ] gives us [ 𝐶 ∗ 1 𝐶 ∗ 0 ] = [ 𝐶 0 ]. This means state 0 has amplitude value of C and state 1 has amplitude value of 0. This is just an example and is not valid in quantum computing because the total probability is less than 1 if C is not 1. If C = 1 only then is it valid. The state vector, mathematically, can be used as a n x 1 matrix to carry out computations. An nx1 matrix is n rows and one column. The reason is that the total probability must equal 1 is that the operations of matrices have to be reversible which means a matrix has to be unitary. The notation for each state vector is called a ket. The ket for state 0 is |0> and the ket for state 1 is |1>. A ket is part of the bra-ket notation. A bra-ket notation is also known as Dirac notation. Paul Dirac was a physicist that came up with the notation for quantum mechanics. Bra is a linear form of the vector, which is a 1xn matrix, meaning one row and n columns. The way we obtain bra from ket is we perform the complex conjugate transpose of the ket form. The representation of bra for state 0 is <0| and bra for state 1 is
- 7. <1|. We use bra to do inner products which looks like <0|0>. The left side is called bra and the right side is called ket. The next article will be about quantum gates that represent quantum states and two qubits. We have so far talked about one qubit. But what do two qubits look like? I will answer this in my next article. I would like to thank the Womanium Global Quantum Computing Program for encouraging me to start sharing my journey into the quantum world. Below are resources, references and futher readings: https://gitlab.com/qworld/bronze-qiskit qiskit.org/learn/course/introduction-course wikipedia https://en.wikipedia.org/wiki/Quantum_computing https://en.wikipedia.org/wiki/ Quantum_technology https://en.wikipedia.org/wiki/DiVincenzo’s_criteria https://en.wikipedia.org/wiki/Quantum_information book: Quantum Computation and Quantum Information by Michael Nielsen and Isaac Chuang (textbook also referred to as Mike & Ike)
- 8. Introduction to quantum mechanics and quantum computing (part 3) Gates In the first article I talked about quantum mechanics. In my second article I talked about quantum computing. In this article I will talk about gates, two qubits and how two qubits look like. We start this talk with the bloch sphere. This is a visual representation of a qubit. There are three axis, the X-axis, Z-axis and Y-axis. We will focus on the X and Z axis because they are straightforward. The Y-axis contains the imaginary plane. A picture of the bloch sphere is shown below and will be referenced in this article and the rest of the series. (Pic) Usually in quantum computing, the qubits start out in state 0 like classical computing the bits start at state 0. Quantum logical gates are called gates and allow us to perform operations mathematically in quantum computing. So, an important and common gate is the X-gate it behaves like a NOT in classical computing. It is a basic one qubit gate. An X-gate flips a bit from state 0 to state 1. The notation for this transition is X|0> = |1>. This is because the position on the bloch sphere rotates around the x axis. The matrix representation of a X-gate is [ 0 1 1 0 ] hadamard, not, 2 qubit amplitude, cnot, more advanced gates are a combo other basic gates mentioned, measurements, ….. I would like to thank the Womanium Global Quantum Computing Program for encouraging me to start sharing my journey into the quantum world. Below are resources, references and futher readings: https://gitlab.com/qworld/bronze-qiskit qiskit.org/learn/course/introduction-course wikipedia https://en.wikipedia.org/wiki/Quantum_computing https://en.wikipedia.org/wiki/ Quantum_technology
- 9. https://en.wikipedia.org/wiki/DiVincenzo’s_criteria https://en.wikipedia.org/wiki/Quantum_information https://en.wikipedia.org/wiki/ https://en.wikipedia.org/wiki/ book: Quantum Computation and Quantum Information by Michael Nielsen and Isaac Chuang (textbook also referred to as Mike & Ike)